Orbit equivalence of global attractors of semilinear parabolic differential equations

We consider global attractors A(f) of dissipative parabolic equations u(t) = u(xx) + f(x, u, u(x)) on the unit interval 0 less than or equal to x less than or equal to 1 with Neumann boundary conditions. A permutation pi(f) is defined by the two ord erings of the set of (hyperbolic) equilibrium solutions u(t) = 0 according to their respective values at the two boundary points x = 0 and x = 1. We p rove that two global attractors, A(f) and A(g), are globally C-0 orbit equi valent, if their equilibrium permutations pi(f) and pi(g) coincide. In othe r words, some discrete information on the ordinary differential equation bo undary value problem u(t) = 0 characterizes the attractor of the above part ial differential equation, globally, up to orbit preserving homeomorphisms.